Re: Relative Cardinality

In article <1121606374.620058.112390@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> Virgil wrote:
>
> > But does such a "number" exist in WM's peculiar worldview?
> > What about the continued fraction represented by
> > (1:2,2,2,2,...)
>
> It exists as an idea, not as a number.
> >
> > The "reality" of every number is at best potential, as none of them are
> > actual in any physical sense.
>
> By its fundamental set or by an n-adic representation a number is
> realized. If you say bang-bang-bang then that is a realization of the
> number 3. If you hurt 4 fingers of your hand, then that is a
> realization of the number 4.

But "fingers" are too indefinite, as they are of unequal sizes and
possibly of unequal completeness. How much of a "finger" must be lost
before it is no longer counts as a finger? None of these "realizations"
has sharp enough boundaries to be only a single number. Only in the
ideal world of the imagination can one imagine a number untrammeled by
the fuzzyness of physicality.
>
>
> > > But not every rational is a natural! You argued that a set of rationals
> > > can contain infinitely many finite elements. I did not argue against
> > > that, but said that an actually infinite set of naturals cannot exist
> > > without an actually infinite element.
> >
> > WRONG! That is like saying that every sequence converges.
>
> It is like saying that every convergent sequence converges.
But WM makes the sequence of naturals converge by giving it a limit.

How is the sequence of rationals (1/1,2/1,3/1,...) different from the
sequence of naturals (1,2,3,...)?

That sequence of rationals exists and contains infinitely many terms but
does not have a limit. According to WM that sequence of naturals is
finite.
.

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